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Monotone normality and compactness. (English) Zbl 0874.54004

The author constructs two spaces which provide negative answers for several well-known questions about monotonically normal spaces.
Example 1. In ZFC, there exists a locally compact monotonically normal space such that none of its compactifications is monotonically normal.
Example 2. In \(\text{ZFC}+ \diamondsuit\), there exists a compact \(K_1\)-space which is not \(K_0\).
Reviewer: C.R.Borges (Davis)

MSC:

54A35 Consistency and independence results in general topology
54E20 Stratifiable spaces, cosmic spaces, etc.
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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References:

[1] Borges, C. J.R., A study of monotonically normal spaces, (Proc. Amer. Math. Soc., 38 (1973)), 211-214 · Zbl 0257.54018
[2] Collins, P. J., Monotone normality, Topology Appl., 74, 179-198 (1996) · Zbl 0867.54027
[3] Heath, R. W.; Lutzer, D. J.; Zenor, P. L., Monotonically normal spaces, Trans. Amer. Math. Soc., 178, 481-493 (1973) · Zbl 0269.54009
[4] J. Nikiel, S. Purisch and L.B. Treybig, Separable, zero-dimensional spaces which are continuous images of ordered compacta, Preprint.; J. Nikiel, S. Purisch and L.B. Treybig, Separable, zero-dimensional spaces which are continuous images of ordered compacta, Preprint. · Zbl 0965.54034
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